Multiple in multiple out network coded amplify and forward relaying scheme for three node bidirectional cooperation

ABSTRACT

A method and apparatus are described including receiving a first signal including first data in a first time slot of a first channel, receiving a second signal including second data in a second time slot of a second channel, determining a first pre-coding matrix, determining a second pre-coding matrix, applying the first pre-coding matrix to the first data to produce pre-coded first data, applying the second pre-coding matrix to the second data to produce pre-coded second data, generating a third signal by combining the pre-coded first data and the pre-coded second data and transmitting the third signal on the first channel and on the second channel. Also described are a method and apparatus including transmitting a first signal, receiving a second signal including a first training sequence and decoding the second signal by removing the first training sequence and removing the first signal.

RELATED APPLICATIONS

This application is a divisional of co-pending U.S. application Ser. No.13/810,408, filed Jan. 15, 2013, which is a 371 of InternationalApplication PCT/US10/43655 filed Jul. 29, 2010 herein incorporated byreference.

FIELD OF THE INVENTION

The present invention is directed to a three node cooperation scheme toassist in bidirectional transmissions (communications) of the draft IEEE802.11n standard.

BACKGROUND OF THE INVENTION

In multicast and broadcast applications, data are transmitted from aserver to multiple receivers over wired and/or wireless networks. Amulticast system as used herein is a system in which a server transmitsthe same data to multiple receivers simultaneously, where the receiversform a subset of all the receivers up to and including all of thereceivers. A broadcast system is a system in which a server transmitsthe same data to all of the receivers simultaneously. That is, amulticast system by definition can include a broadcast system.

Consider multicast (downlink) and multi-access (uplink) channels withone access point (AP) and several nodes. In the IEEE 802.11n draftstandard, a reverse direction (RD) protocol is introduced for fastscheduling of bidirectional traffic flows within a transmissionopportunity (TXOP). The reverse direction protocol permits (allows) thenode, which has obtained the TXOP to grant reverse directionaltransmissions to another node while it is still in control of the TXOP.If the channel conditions between the nodes are inadequate (poor) thentransmissions between the two nodes suffer. That suffering may bereduced data rate and/or throughput.

In the IEEE 802.11n draft standard, a reverse direction (RD) protocolhas been proposed as in FIG. 1. The reverse direction protocol of theIEEE 802.11n draft standard only schedules bidirectional transmissionbetween two nodes. Each node is both a source node and a destinationnode. There is no existing scheduling protocol for three-nodebidirectional transmissions in IEEE 802.11 WLAN standards. FIG. 1illustrates the conventional unidirectional cooperation using ahalf-duplex relay node (RN). FIG. 1 a shows the first stage ofcommunication, in which Node₁ transmits (sends, communicate) data S₁ toboth Node₂ and the RN. FIG. 1 b shows stage 2 of communication, in whichthe RN transmits (communicates, sends) data S₁ to Node₂. That is, the RNtransmits (communicates, forwards, sends) data S₁as Ŝ₁ to Node₂.Correspondingly (and not shown), in the third stage of communication,Node₂ transmits (sends, communicate) data S₂ to both Node₁ and the RN.In the fourth stage of communication, the RN transmits (communicates,forwards, sends) data Ŝ₂ to Node₁. That is, the RN transmits(communicates, forwards, sends) data S₂ as Ŝ₂ to Node₁. Thus, in theconventional approach, there are four stages (phases) to completecommunications using the half-duplex RN to assist Node₁ and Node₂.

Network-coded three-node bidirectional cooperation with three stages(i.e., the reception of signals from nodes (source and destination) atthe RN are orthogonal (separate)) has been studied, usingDecode-and-Forward, Soft Decode-and-Forward, and Amplify-and-Forward ina single-antenna system, and a case when L₁=L₂=1 and L_(R)=2 etc,respectively. Note that L_(i), i=1,2,R represents the number of antennasat Node₁, Node₂ and RN respectively. The present invention usesAmplify-and-Forward for the general MIMO case with an arbitrary numberof antennas at the nodes. This has not been addressed in anypublications known to the Applicants.

SUMMARY OF THE INVENTION

As used herein a node includes (but is not limited to) a station (STA),a mobile device, a mobile terminal, a dual mode smart phone, a computer,a laptop computer or any other equivalent device capable of operatingunder the IEEE 802.11n draft standard.

Consider multicast (downlink) and multi-access (uplink) channels withone access point (AP) and several nodes. In the IEEE 802.11n draftstandard, a reverse direction (RD) protocol is introduced for fastscheduling of bidirectional traffic flows within a transmissionopportunity (TXOP). The reverse direction protocol permits (allows) thenode, which has obtained the TXOP to grant reverse directionaltransmissions to another node while it is still in control of the TXOP.However, when the channel state (channel conditions) between the twonodes is not sufficient for providing fast and reliable transmissions(communication) between them, cooperation through a third node, ahalf-duplex relay node (RN), may be involved to assist transmissions(communications). When the transmission between these two nodes involvescooperation through a third node, a half-duplex relay node (RN), thesituation becomes more complicated, and wireless network coding can beutilized to further increase system throughput. Each node is both asource node and a destination node.

In the present invention, as shown in FIG. 2 and described more fullybelow, wireless network coding is introduced into the system andcombined with bidirectional cooperation to increase system throughput.The present invention describes a network-coded amplify-and-forward(NCAF) relaying scheme in such a three-node bidirectional cooperationscenario.

In three-node bidirectional cooperation scheme of the present invention,two nodes, Node₁ and Node₂, are both source and destination nodes, andthe RN is a relay node, assisting the bidirectional transmission betweenNode₁ and Node₂. The relay node (RN) sequentially receives signals fromboth Node₁ and Node₂, combines the two signals with pre-coding matricesfor both of them, and broadcasts the mixed signal to both nodes onorthogonal channels. Each of the nodes (source and destination) receivesboth the transmission (communication) of the desired signal from theother node, and the transmission (communication) of the mixed signalfrom the RN. Each node can jointly decode the data it desires based onthe knowledge of the signal it sent out (transmitted, communicated). Theprocess is shown in FIG. 3, which will be further described below. Thepresent invention not only describes the above network-codedamplify-and-forward (NCAF) relaying scheme, but also solves the designproblem of the pre-coding matrices at RN for the received signals fromthe nodes (source and destination), so as to maximize the instantaneouscapacity of the bidirectional cooperation system, subject to a totalpower constraint at the RN, given channel state information (CSI) in twocases:

(1) Without CSI of direct links at the RN: Only CSI of the channels fromthe nodes (source and destination) to the RN, and the channels from theRN to the nodes (source and destination), are known at the RN. CSI ofthe channels between the two nodes (source and destination) is not knownat the RN.

(2) With CSI of the direct links at the RN: CSI of the channels from thenodes (source and destination) to the RN, and the channels from RN tothe nodes (source and destination), and the channels between the twonodes (source and destination) are known at the RN.

In the Network-Coded Amplify-and-Forward (NCAF) relaying scheme of thepresent invention, the RN no longer forwards the amplified receivedsignals from one node to another as in conventional amplify-and-forwardcooperation. Instead, the RN combines two received signals from twonodes by firstly multiplying them with pre-coding matrices. Then the RNbroadcasts (multicasts) the combined signal, which contains the mixeddata of the bidirectional traffic flow. Each end node receives thesignal from the RN. The nodes can then jointly decode its desired signalbased on the knowledge of the signal it has sent out (transmitted,communicated). There is still diversity in the cooperation.

The NCAF relaying scheme of the present invention, may prove to beessential to the future the IEEE 802.11 draft Very High Throughput (VHT)standard. The advantage of the NCAF relaying scheme of the presentinvention is that only simple processing, i.e., linear pre-coding, isneeded at the relay node (RN). It is also compatible with conventionalcooperation using an amplify-and-forward relaying scheme. The NCAFrelaying scheme of the present invention also solves the problem whenthe RN is not able to decode the received data due to an insufficientnumber of antennas equipped at the RN and is always feasible for anymultiple-antenna systems.

A method and apparatus are described including receiving a first signalincluding first data in a first time slot of a first channel, receivinga second signal including second data in a second time slot of a secondchannel, determining a first pre-coding matrix, determining a secondpre-coding matrix, applying the first pre-coding matrix to the firstdata to produce pre-coded first data, applying the second pre-codingmatrix to the second data to produce second pre-coded data, generating athird signal by combining the pre-coded first data and the pre-codedsecond data and transmitting the third signal on the first channel andon the second channel. Also described are a method and apparatusincluding transmitting a first signal, receiving a second signal andjointly decoding the second signal including a first training sequenceby removing the first training sequence and removing the first signal.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is best understood from the following detaileddescription when read in conjunction with the accompanying drawings. Thedrawings include the following figures briefly described below:

FIG. 1A shows the first stage of communication, in which Node₁ transmits(sends, communicate) data S₁ to both Node₂ and the RN.

FIG. 1B shows stage 2 of communication, in which the RN transmits(communicates sends) data S₁ to Node₂.

FIG. 2A shows the first stage of communication of the present invention,in which Node₁ transmits (sends, communicates) data S₁ to both Node₂ andthe RN.

FIG. 2B shows the second stage of communication of the presentinvention, in which Node₂ transmits (sends, communicate) data S₂ to bothNode and the RN.

FIG. 2C shows the third stage of communication of the present invention,in which the RN combines (mixes) data Ŝ₁+Ŝ₂ for transmission(communication) to both Node₁ and Node₂.

FIG. 3A is a block diagram of the operation of an exemplary embodimentof the transmission side of the Network-Coded Amplify-and-Forwardrelaying scheme of the present invention.

FIG. 3B is a block diagram of the operation of an exemplary embodimentof the reception side of the Network-Coded Amplify-and-Forward relayingscheme of the present invention from the perspective of Node₁.

FIG. 3C is a block diagram of the operation of an exemplary embodimentof the reception side of the Network-Coded Amplify-and-Forward relayingscheme of the present invention from the perspective of Node₂.

FIG. 4A shows the RN applying the pre-coding matrices to the trainingsequences it receives and forwarding them.

FIG. 4B shows the RN estimating the incoming channel matrices,multiplying them with the pre-coding matrices, quantizing the resultingmatrices, and feeding them back. The RN also sends out its own trainingsequence to the nodes (source and destination) to estimate the channelstate (channel conditions) from the RN to them.

FIG. 5A shows the RN applying the pre-coding matrices to the trainingsequences it receives and forwarding them.

FIG. 5B shows the RN estimating the incoming channel matrices,multiplying them with the pre-coding matrices, quantizing the resultingmatrices, and feeding them back. The RN also sends out its own trainingsequence to the nodes (source and destination) to estimate the channelstate (channel conditions) from the RN to them.

FIG. 6 is a flowchart of an exemplary embodiment of the presentinvention from the perspective of a node (source and destination).

FIG. 7 is a flowchart of an exemplary embodiment of the presentinvention from the perspective of a relay node.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 2 shows the operation of the Network-Coded Amplify-and-Forwardrelaying scheme for bidirectional cooperation of the present inventionusing a half-duplex relay node. FIG. 2 a shows the first stage ofcommunication, in which Node₁ transmits (sends, communicates) data S₁ toboth Node₂ and the RN. In the second stage of communication as shown inFIG. 2 b, Node₂ transmits (sends, communicate) data S₂ to both Node₁ andthe RN. In the third stage of communication as shown in FIG. 2 c, the RNcombines (mixes) data Ŝ₁+Ŝ₂ for transmission (communication) to bothNode₁ and Node₂. Data Ŝ₁+Ŝ₂ is formed by applying pre-coding matrices todata S₁ and data S₂. The combination of data Ŝ₁+Ŝ₂ is f(S₁+S₂). Thenumber of stages has thus been reduced from four stages to three stages.It should be noted that in addition to applying pre-coding matrices tothe data, beamforming matrices may also be applied to the data.

A system model and some notation is introduced first. X_(i) is thesignal transmitted from Node, (i=1,2,R, and Node_(R) denotes the RN).Q_(i) is the beamforming matrix at Node_(i), and it is an identitymatrix if beamforming is not used. H_(ij) is the channel matrix fromNode_(j) to Node_(i). Y_(ij) and N_(ij) are the received signal andnoise from Node_(j) to Node_(i), elements are assumed to be independentand identically distributed Gaussian with variance σ_(n) ². W_(i) is thepre-coding matrix applied to Y_(Ri) at RN. The total transmitted poweris P_(i) at Node_(i) (i=1,2) and P_(R) at the RN.

The RN receives Y_(R1)=H_(R1)Q₁X₁+N_(R1) from Node₁ in the 1^(st) timeslot, and Y_(R2)=H_(R2)Q₂X₂+N_(R2) from Node₂ in the 2^(nd) time slot.Then the RN mixes the weighted received signals as its broadcast signal

X _(R) =W ₁ Y _(R1) +W ₂ Y _(R2) =W ₁ H _(R1) Q ₁ X ₁ +W ₂ H _(R2) Q ₂ X₂ +W ₁ N _(R1) +W ₂ N _(R2).

The reception and decoding at one node (source and destination) is usedas an example. It goes without saying that demodulation is performed atthe reception end. The processing at the other node (source anddestination) is similar. Node₁ receives Y₁₂=H₁₂Q₂X₂+N₁₂ from Node₂ inthe 2^(nd) time slot, and Y_(1R)=H_(1R)X_(R)+N_(1R) from the RN in the3^(rd) time slot, i.e.,Y_(1R)=H_(1R)W₁H_(R1)Q₁X₁+H_(1R)W₂H_(R2)Q₂X₂+H_(1R)W₁N_(R1)+H_(1R)W₂N_(R2)+N_(1R).The matrix form is

$\begin{matrix}{{\begin{bmatrix}Y_{12} \\Y_{1\; R}\end{bmatrix} = {{\begin{bmatrix}0_{N_{1}} \\{H_{1R}W_{1}H_{R\; 1}}\end{bmatrix}Q_{1}X_{1}} + {\begin{bmatrix}H_{12} \\{H_{1R}W_{2}H_{R\; 2}}\end{bmatrix}Q_{2}X_{2}} + {\begin{bmatrix}I_{N_{1}} & 0_{N_{1} \times N_{R}} & 0_{\; {N_{1} \times N_{R}}} & 0_{N_{1}} \\0_{N_{1}} & {H_{1R}W_{1}} & {H_{1R}W_{2}} & I_{N_{1}}\end{bmatrix}\begin{bmatrix}N_{12} \\N_{R\; 1} \\N_{R\; 2} \\N_{1R}\end{bmatrix}}}},} & (1)\end{matrix}$

i.e., Y₁=D₁X₁+A₁X₂+B₁N₁, where Y₁=[Y₁₂ Y_(1R)]^(T)∈C^(2L) ¹ ^(×1), 0_(N)_(i) _(×N) _(j) ∈C^(L) ^(i) ^(×L) ^(j) is a zero matrix, andI_(N)∈C^(N×N) is an identity matrix. Note that at Node₁, X₁ is known,but X₂ is unknown and to be detected. N₁=[N₁₂ N_(R1) N_(R2)N_(1R)]^(T)∈C^(2(L) ¹ ^(+L) ^(R) ^()×1) is the noise vector at Node₁,and the matrices,

${D_{1} = {\begin{bmatrix}0_{N_{1}} \\{H_{1R}W_{1}H_{R\; 1}}\end{bmatrix}Q_{1}}},{A_{1} = {\begin{bmatrix}H_{12} \\{H_{1R}W_{2}H_{R\; 2}}\end{bmatrix}Q_{2}{\mspace{11mu} \;}{and}}}$$B_{1} = \begin{bmatrix}I_{N_{1}} & 0_{N_{1} \times N_{R}} & 0_{\; {N_{1} \times N_{R}}} & 0_{N_{1}} \\0_{N_{1}} & {H_{1R}W_{1}} & {H_{1R}W_{2}} & I_{N_{1}}\end{bmatrix}$

are assumed known. Based on the knowledge of X₁, Node₁ can obtainZ₁=Y₁−D₁X₁=A₁X₂+B₁N₁, where Z₁ is the equivalent received signal, and A₁is the equivalent channel matrix for signal X₂. Then Node₁ can jointlydecode X₂.

Similarly, Z₂=A₂X₁+B₂N₂, where Z₂, A₂, B₂ and N₂ are defined byexchanging the subscripts “1” and “2”, and “N₁” and “N₂” in theircounterparts.

The problem is to determine W₁ and W₂, W_(i)∈C^(L) ^(R) ^(×L) ^(R) ,i=1,2, to maximize the instantaneous capacity of the system, subject tothe transmit power constraint at the RN. W₁ and W₂ are the pre-codingmatrices. That is, determine W₁ and W₂, to maximize

$\begin{matrix}{{f = {{\log \; {\det \left( {I_{2N_{1}} + {\frac{P_{2}}{N_{2}\sigma_{n}^{2}}\left( {A_{1}A_{1}^{H}} \right)\left( {B_{1}B_{1}^{H}} \right)^{- 1}}} \right)}} + {\log \; {\det \left( {I_{2N_{2}} + {\frac{P_{1}}{N_{1}\sigma_{n}^{2}}\left( {A_{2}A_{2}^{H}} \right)\left( {B_{2}B_{2}^{H}} \right)^{- 1}}} \right)}}}},{{{{subject}\mspace{14mu} {to}\mspace{14mu} \frac{P_{1}}{N_{1}}{tr}\left\{ {W_{1}H_{R\; 1}Q_{1}Q_{1}^{H}H_{R\; 1}^{H}W_{1}^{H}} \right\}} + {\sigma_{n}^{2}{tr}\left\{ {W_{1}W_{1}^{H}} \right\}} + {\frac{P_{2}}{N_{2}}{tr}\left\{ {W_{2}H_{R\; 2}Q_{2}Q_{2}^{H}H_{R\; 2}^{H}W_{2}^{H}} \right\}} + {\sigma_{n}^{2}{tr}\left\{ {W_{2}W_{2}^{H}} \right\}}} = P_{R}},} & (1)\end{matrix}$

where tr(X) represents the trace of the matrix X.Letting Q_(i), i=1,2 be a unitary matrix, the constraint can besimplified to

$\begin{matrix}{{{\frac{P_{1}}{N_{1}}{tr}\left\{ {W_{1}H_{R\; 1}H_{R\; 1}^{H}W_{1}^{H}} \right\}} + {\sigma_{n}^{2}{tr}\left\{ {W_{1}W_{1}^{H}} \right\}} + {\frac{P_{2}}{N_{2}}{tr}\left\{ {W_{2}H_{R\; 2}H_{R\; 2}^{H}W_{2}^{H}} \right\}} + {\sigma_{n}^{2}{tr}\left\{ {W_{2}W_{2}^{H}} \right\}}} = {P_{R}.}} & (2)\end{matrix}$

In the first case, the CSI of the channels (links) from Node_(i) to theRN and the RN to Node_(j), are assumed available at the RN. The CSI ofthe links (channels) between Node_(i) and Node_(j), i,j=1,2 and i≠j isnot available. In this scenario, it is necessary to maximize an upperbound of f instead of f itself due to the lack of information in it.That is, determine W₁ and W₂, to maximize

$\begin{matrix}{{f_{1} = {{\log \; {\det \left( {I_{N_{1}} + {\frac{P_{2}}{N_{2}\sigma_{n}^{2}}H_{1R}W_{2}H_{R\; 2}H_{R\; 2}^{H}W_{2}^{H}{H_{1R}^{H}\left( {I_{N_{1}} + {H_{1R}W_{1}W_{1}^{H}H_{1R}^{H}} + {H_{1R}W_{2}W_{2}^{H}H_{1R}^{H}}} \right)}^{- 1}}} \right)}} + {\log \; {\det \left( {I_{N_{2}} + {\frac{P_{1}}{N_{1}\sigma_{n}^{2}}H_{2R}W_{1}H_{R\; 1}H_{R\; 1}^{H}W_{1}^{H}{H_{2R}^{H}\left( {I_{N_{2}} + {H_{2R}W_{1}W_{1}^{H}H_{2R}^{H}} + {H_{2R}W_{2}W_{2}^{H}H_{2R}^{H}}} \right)}^{- 1}}} \right)}}}}{{{{subject}\mspace{14mu} {to}\mspace{14mu} \frac{P_{1}}{N_{1}}{tr}\left\{ {W_{1}H_{R\; 1}H_{R\; 1}^{H}W_{1}^{H}} \right\}} + {\sigma_{n}^{2}{tr}\left\{ {W_{1}W_{1}^{H}} \right\}} + {\frac{P_{2}}{N_{2}}{tr}\left\{ {W_{2}H_{R\; 2}H_{R\; 2}^{H}W_{2}^{H}} \right\}} + {\sigma_{n}^{2}{tr}\left\{ {W_{2}W_{2}^{H}} \right\}}} = P_{R}}} & (3)\end{matrix}$

as shown in Equation (2).

In the second case, the CSI of channels from Node, to the RN, from theRN to Node_(j), and from Node_(i) to Node_(j), are assumed to beavailable to the RN, i,j≠1,2 and i≠j. In this scenario, the designproblem is to maximize (1) subject to (2).

By the singular value decomposition (SVD) theorem, the channels (links)can be decomposed to H_(R1)=U_(R1)A_(R) ^(1/2)V_(R1) ^(H),H_(R2)=U_(R2)A_(R2) ^(1/2)V_(R2) ^(H), H_(1R)=U_(1R)A_(1R) ^(1/2)V_(1R)^(H) and H_(2R)=U_(2R)A_(2R) ^(1/2)V_(2R) ^(H, where U)_(R1),U_(R2),V_(1R),V_(2R)∈C^(L) ^(R) ^(×L) ^(R) , V_(R1),U_(1R)∈C^(L) ¹^(×L) ¹ , and V_(R2),U_(2R)∈C^(L) ² ^(×L) ² are unity matrices; A_(ij)^(1/2)∈C^(L) ^(i) ^(×L) ^(j) , i, j=1,2,R, are singular value matrices.In particular, A_(ij)=A_(ij) ^(1/2)(A_(ij)^(1/2))^(H)=diag{λ_(ij,k)}∈C^(L) ^(i) ^(×L) ^(i) , i, j=1,2,R, where()^(T) and ()^(H) represent the matrix transpose and conjugatetranspose operations, respectively. Also define T=V_(1R) ^(H)_(R)V_(2R)∈C^(L) ^(R) ^(×L) ^(R) and denote T=(t_(ij)), i, j=1, . . . ,L_(R) .

When the CSI of direct links between Node₁ and Node₂ is not available,let W₁=V_(2R){tilde over (W)}₁U_(R1) ^(H) and W₂=V_(1R){tilde over(W)}₂U_(R2) ^(H), where {tilde over (W)}₁, {tilde over (W)}₂∈C^(L) ^(R)^(×L) ^(R) are to be determined. There is no closed form solution. But{tilde over (W)}_(l) and {tilde over (W)}₂ can be solved iterativelyusing Newton's method.

Furthermore, the problem statement can be rewritten in the followingform, where the notation will be explained after the approach isintroduced.

Determine λ, to minimize f₂(λ)   (4)

subject to λ≧0,   (5)

Sλ=q.   (6)

The Lagrangian function is L(λ, μ)=f ₂(λ) −μ^(T)(Sλ−q),   (7)

where μ, is the vector containing the Lagrangian multipliers. UsingNewton's method, the following iterative approach is used to solve forλ:

Step 1: Initiate λ⁰∈(0,max_λ).

Step 2: In each iteration, solve

$\begin{matrix}{{{\begin{bmatrix}{\nabla^{2}{f_{2}\left( \lambda^{k} \right)}} & {- S^{T}} \\{- S} & 0\end{bmatrix}v^{k}} = \begin{bmatrix}{S^{T}\mu^{k}} & {- {\nabla{f_{2}\left( \lambda^{k} \right)}}} \\{S\; \lambda^{k}} & {- q}\end{bmatrix}}{{{for}\mspace{14mu} v^{k}} = {\begin{pmatrix}v_{\lambda}^{k} \\v_{\mu}^{k}\end{pmatrix}.}}} & (8)\end{matrix}$

Step 3: Take as next iteration

$\begin{matrix}{\begin{pmatrix}\lambda^{k + 1} \\\mu^{k + 1}\end{pmatrix} = {\begin{pmatrix}\lambda^{k} \\\mu^{k}\end{pmatrix} + {\begin{pmatrix}v_{\lambda}^{k} \\v_{\mu}^{k}\end{pmatrix}.}}} & (9)\end{matrix}$

Step 4: If sum(sign(λ^(k+1)))≠length(λ^(k+1)) orsum(sign(max_λ−λ^(k+1)))≠length(λ^(k+1)), go back to Step 1. Otherwise,go to Step 5.

Step 5: If ∥v_(λ) ^(k)∥²<Threshold, stop. Otherwise, k=k+1, and go toStep 2.

In Step 4, given x=(x₁, . . . , x_(M))^(T) is a column vector or lengthM, length(x)=M and

${{sum}(x)} = {{\sum\limits_{i = 1}^{{length}{(x)}}{x_{i} \cdot {{sign}(x)}}} = \left\{ {\begin{matrix}{1,} & {{{if}\mspace{14mu} x} > 0} \\{0,} & {{{if}\mspace{14mu} x} = 0} \\{{- 1},} & {{{if}\mspace{11mu} x} < 0}\end{matrix},{{{and}\mspace{14mu} {{sign}(x)}} = {\left( {{{sign}\left( x_{1} \right)},\ldots \;,{{sign}\left( x_{M} \right)}} \right)^{T}.}}} \right.}$

Condition α₁ is defined as “when L_(i)≧L_(R), or when both L_(i)<L_(R)and L_(i)(L_(i)−1)≧L_(R) are satisfied”, and condition β_(i) is definedas “when both L_(i)<L_(R) and L_(i)(L_(i)−1)≦L_(R)−1 are satisfied”.Therefore, there are three cases of solution as follows:

General Case 1: When α₁ and α₂ exist, both {tilde over (W)}₁ and {tildeover (W)}₂ are identity matrices multiplied by constants, respectively,i.e., {tilde over (W)}₁=λ₁I_(L) _(R) , {tilde over (W)}₂=λ₂I_(L) _(R) ,λ₁, λ₂≧0. The notation in the iterative approach is as follows:

${\lambda = \left( {\lambda_{1},\lambda_{2}} \right)^{T}},{{max\_\lambda} = \left( {{P_{R}\text{/}d_{a}\sigma_{n}^{2}},{P_{R}\text{/}d_{b}\sigma_{n}^{2}}} \right)^{T}},{µ = {{\mu \mspace{14mu} {and}\mspace{14mu} v^{k}} = {{\left( {v_{\lambda_{1}}^{k},v_{\lambda_{2}}^{k},v_{\mu}^{k}} \right)^{T}.S} = \left( {d_{a},d_{b}} \right)}}},{q = {P_{R}\text{/}\sigma_{n}^{2}}},{{{where}\mspace{14mu} c_{a,i}} = {{\frac{P_{1}}{L_{2}\sigma_{n}^{2}}\lambda_{{R\; 1},i}} + 1}},{c_{b,i} = {{\frac{P_{2}}{L_{2}\sigma_{n}^{2}}\lambda_{{R\; 2},i}} + 1}},{i = 1},\ldots \;,M,{d_{a} = {\sum\limits_{i = 1}^{M}c_{a,i}}},$

${d_{b} = {\sum\limits_{i = 1}^{M}c_{b,i}}},$

and M=min, (L₁, L₂, L_(R)). The Hessian and gradient of f₂(λ) areexpressed as following:

${{\nabla^{2}{f_{2}\left( \lambda^{k} \right)}} = {\begin{pmatrix}\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda^{2}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1}}{\partial\lambda_{2}}} \\\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1}}{\partial\lambda_{2}}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2}^{2}}\end{pmatrix}\mspace{14mu} {and}}}\;$${{\nabla{f_{2}\left( \lambda^{k} \right)}} = \begin{pmatrix}\frac{\partial{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{1}} \\\frac{\partial{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2}}\end{pmatrix}}\;,{where}$${\frac{\partial{f_{2}(\lambda)}}{\partial\lambda_{1}} = {\sum\limits_{i = 1}^{M}\left( {{- m_{1,i}} + m_{2,i} - {c_{a,i}m_{3,i}} + m_{4,i}} \right)}},{\frac{\partial{f_{2}(\lambda)}}{\partial\lambda_{2}} = {\sum\limits_{i = 1}^{M}\left( {{{- c_{b,i}}m_{1,i}} + m_{2,i} - m_{3,i} + m_{4,i}} \right)}},{\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{1}^{2}} = {\sum\limits_{i = 1}^{M}\left( {m_{1,i}^{2} - m_{2,i}^{2} + {c_{a,i}^{2}m_{3,i}^{2}} - m_{4,i}^{2}} \right)}},{\frac{\partial^{2}{f_{2}(\lambda)}}{{\partial\lambda_{1}}{\partial\lambda_{2}}} = {\sum\limits_{i = 1}^{M}\left( {{c_{b,i}m_{1,i}^{2}} - m_{2,i}^{2} + {c_{a,i}m_{3,i}^{2}} - m_{4,i}^{2}} \right)}},{\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{2}^{2}} = {\sum\limits_{i = 1}^{M}\left( {{c_{b,i}^{2}m_{1,i}^{2}} - m_{2,i}^{2} + m_{3,i}^{2} - m_{4,i}^{2}} \right)}},{and}$${m_{1,i} = \frac{\lambda_{{1R},i}}{1 + {\lambda_{1}\lambda_{{1R},i}} + {\lambda_{2}c_{b,i}\lambda_{{1R},i}}}},{m_{2,i} = \frac{\lambda_{{1R},i}}{1 + {\lambda_{1}\lambda_{{1R},i}} + {\lambda_{2}\lambda_{{1R},i}}}},{m_{3,i} = \frac{\lambda_{{2R},i}}{1 + {\lambda_{1}c_{a,i}\lambda_{{2R},i}} + {\lambda_{2}\lambda_{{2R},i}}}},{m_{4,i} = {\frac{\lambda_{{2R},i}}{1 + {\lambda_{1}\lambda_{{2R},i}} + {\lambda_{2}\lambda_{{2R},i}}}.}}$

General Case 2: When α_(l) and β₂ exist, {tilde over (W)}₁ is anidentity matrix multiplied by a constant, i.e., {tilde over (W)}=λ₁I_(L)_(R) , λ₁≧0; while {tilde over (W)}₂=A₂ ^(1/2)=diag{λ_(2,1) ^(1/2), . .. , λ_(2,L) _(R) ^(1/2)}, λ_(2,i)≧0, i=1, . . . , L_(R), is a diagonalmatrix. The notation in the iterative approach is as follows:

${\lambda = \left( {\lambda_{1},\lambda_{2,1},\ldots \;,\lambda_{2,L_{R}}} \right)^{T}},{{max\_\lambda} = \left( {{P_{R}\text{/}d_{a\;}\sigma_{n}^{2}},{P_{R}\text{/}c_{b,1}\sigma_{n}^{2}},L,{P_{R}\text{/}c_{b,M}\sigma_{n}^{2}},1_{L_{R} - M}^{T}} \right)^{T}},{µ = \left( {\mu_{1},L,\mu_{{L_{2}{({L_{2} - 1})}} + 1}} \right)^{T}},{v^{k} = \left( {v_{\lambda}^{k},v_{\lambda_{2,1}}^{k},L,v_{\lambda_{2,L_{R}}}^{k},v_{\mu_{1}}^{k},L,v_{\mu_{{L_{2}{({L_{2} - 1})}} - 1}}^{k}} \right)^{T}},{1_{N} = {{\left( \underset{N}{\underset{}{1,\ldots \;,1}} \right).S} = \begin{bmatrix}0_{{L_{2}{({L_{2} - 1})}} \times 1} & S_{2} \\d_{a} & {c_{b,1},L,c_{b,M},0_{1 \times {({L_{R} - M})}}}\end{bmatrix}}},{q = \begin{pmatrix}0_{{L_{2}{({L_{2} - 1})}} \times 1} \\{P_{R}/\sigma_{n}^{2}}\end{pmatrix}},{{{where}\mspace{14mu} c_{a,i}} = {{\frac{P_{1}}{L_{1}\sigma_{n}^{2}}\lambda_{{R\; 1},i}} + 1}},{c_{b,i} = {{\frac{P_{2}}{L_{2}\sigma_{n}^{2}}\lambda_{{R\; 2},i}} + 1}},{i = 1},L,M,{d_{a} = {\sum\limits_{i = 1}^{M}c_{a,i}}},$

and M=min(L₁, L₂, L_(R)).

And

${S_{2} = {\begin{bmatrix}{{Re}\left\{ s_{2} \right\}} \\{{Im}\left\{ s_{2} \right\}}\end{bmatrix} \in R^{{L_{2}{({L_{2} - 1})}} \times L_{R}}}},$

where Re{·} and Im{·} are the functions which take the real andimaginary parts of the variables, and s₂ comes from the linear equations

${{\sum\limits_{k = 1}^{L_{R}}{\lambda_{2,k}t_{ki}^{*}t_{kj}}} = 0},$

1≦i<j≦L₂, i.e.,

$\mspace{20mu} {{s_{2} = {{\begin{pmatrix}{t_{11}^{*}t_{12}} & {t_{21}^{*}t_{22}} & L & {t_{L_{R} - 1}^{*}t_{L_{R},2}} \\{t_{11}^{*}t_{13}} & {t_{21}^{*}t_{23}} & L & {t_{L_{R} - 1}^{*}t_{L_{R},3}} \\M & M & O & M \\{t_{11}^{*}t_{1,L_{2}}} & {t_{21}^{*}t_{2,L_{2}}} & L & {t_{L_{R} - 1}^{*}t_{L_{R},2}} \\{t_{12}^{*}t_{13}} & {t_{22}^{*}t_{23}} & L & {t_{L_{R},2}^{*}t_{L_{R},3}} \\M & M & O & M \\{t_{12}^{*}t_{1,L_{2}}} & {t_{22}^{*}t_{2,L_{2}}} & L & {t_{L_{R},2}^{*}t_{L_{R},L_{2}}} \\M & M & O & M \\{t_{1,{L_{2} - 1}}^{*}t_{1,L_{2}}} & {t_{2,{L_{2} - 1}}^{*}t_{2,L_{2}}} & L & {t_{L_{R},{L_{2} - 1}}^{*}t_{L_{R},L_{2}}}\end{pmatrix} \in {C^{\frac{L_{2}{({L_{2} - 1})}}{2} \times L_{R}}.\mspace{20mu} {\nabla^{2}{f_{2}\left( \lambda^{k} \right)}}}} = {\begin{pmatrix}\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{1}^{2}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1}}{\partial\lambda_{2,1}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1}}{\partial\lambda_{2,L_{R}}}} \\\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1}}{\partial\lambda_{2,1}}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2,1}^{2}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{2,1}}{\partial\lambda_{2,L_{R}}}} \\M & M & O & M \\\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1}}{\partial\lambda_{2,L_{R}}}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{2,1}}{\partial\lambda_{2,1}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2,L_{R}}^{2}}\end{pmatrix}\mspace{14mu} {and}}}},\mspace{20mu} {{{where}\mspace{14mu} \frac{\partial f_{2}}{\partial\lambda_{1}}} = {\sum\limits_{i = 1}^{M}\left( {{- m_{1,i}} + m_{2,i} - {c_{a,i}m_{3,i}} + m_{4,i}} \right)}},\mspace{20mu} {\frac{\partial{f_{2}(\lambda)}}{\partial\lambda_{2,j}} = \left\{ {\begin{matrix}{{{{- c_{b,j}}m_{1,j}} + m_{2,j} + {\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {{- m_{3,i}} + m_{4,i}} \right)}}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {{- m_{3,i}} + m_{4,i}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{1}^{2}} = {\sum\limits_{i = 1}^{M}\left( {m_{1,i}^{2} - m_{2,i}^{2} + {c_{a,i}^{2}m_{3,i}^{2}} - m_{4,i}^{2}} \right)}},{\frac{\partial^{2}{f_{2}(\lambda)}}{{\partial\lambda_{1}}{\partial\lambda_{2,j}}} = \left\{ {\begin{matrix}{{{c_{b,j}m_{1,j}^{2}} - m_{2,j}^{2} + {\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {{c_{a,i}m_{3,i}^{2}} - m_{4,i}^{2}} \right)}}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {{c_{a,i}m_{3,i}^{2}} - m_{4,i}^{2}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{2,j}^{2}} = \left\{ {\begin{matrix}{{{c_{b,j}^{2}m_{1,j}^{2}} - m_{2,j}^{2} + {\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {m_{3,i}^{2} - m_{4,i}^{2}} \right)}}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ji}}^{4}\left( {m_{3,i}^{2} - m_{4,i}^{2}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{{\partial\lambda_{2,j}}{\partial\lambda_{2,l}}} = {\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}{t_{li}}^{2}\left( {m_{3,i}^{2} - m_{4,i}^{2}} \right)}}},{1 \leq j},{l \leq L_{R}},{j \neq l},\mspace{20mu} {m_{1,i} = \frac{\lambda_{{1R},i}}{1 + {\lambda_{1}\lambda_{{1\; R},i}} + {\lambda_{2}c_{b,i}\lambda_{{1R},i}}}},{m_{2,i} = \frac{\lambda_{{1R},i}}{1 + {\lambda_{1}\lambda_{{1R},i}} + {\lambda_{2}\lambda_{{1R},i}}}},\mspace{20mu} {m_{3,i} = \frac{\lambda_{{2R},i}}{1 + {\lambda_{1}c_{a,i}\lambda_{{2R},i}} + {\sum\limits_{k = 1}^{L_{R}}{\lambda_{2,k}{t_{ki}}^{2}\lambda_{{2R},i}}}}},\mspace{20mu} {m_{4,i} = {\frac{\lambda_{{2R},i}}{1 + {\lambda_{1}\lambda_{{2R},i}} + {\sum\limits_{k = 1}^{L_{R}}{\lambda_{2,k}{t_{ki}}^{2}\lambda_{{2R},i}}}}.}}} \right.}} \right.}} \right.}}$

General Case 3: When β₁ and β₂ exist, both W₁ ^(%)=A₁ ^(1/2)={λ_(1,1)^(1/2),L,λ_(1,L) _(R) ^(1/2)}, λ_(1,i)≧0, i=1, L, L_(R), and W₂ ^(%)=A₂^(1/2)=diag{λ_(2,1) ^(1/2),L,λ_(2,L) _(R) }, λ_(2,i)≧0, i=1, L, L_(R),are diagonal matrices. The notation in the iterative approach is asfollows:

$\mspace{20mu} {{\lambda = \left( {\lambda_{1,1},{L\; \lambda_{1,N_{R}}},\lambda_{2,1},{L\mspace{11mu} \lambda_{2,L_{R}}}} \right)^{T}},\mspace{20mu} {µ = \left( {\mu_{1},L,\mu_{{L_{2}{({L_{2} - 1})}} + 1}} \right)^{T}},{{max\_\lambda} = \left( {{P_{R}\text{/}c_{a,1}\sigma_{n}^{2}},L,{P_{R}\text{/}c_{a,M}\sigma_{n}^{2}},1_{L_{R} - M}^{T},{P_{R}\text{/}c_{b,1}\sigma_{n}^{2}},L,{P_{R}\text{/}c_{b,M}\sigma_{n}^{2}},1_{L_{R} - M}^{T}} \right)^{T}},{1_{N} = \left( {1,{\ldots_{\begin{matrix}\; \\N\end{matrix}}\ldots},1} \right)},{{{and}\mspace{14mu} v^{k}} = {{\left( {v_{\lambda_{1,1}}^{k},L,v_{\lambda_{1,L_{R}}}^{k},v_{\lambda_{2,1}}^{k},L,v_{\lambda_{2,L_{R}}}^{k},v_{\mu_{1}}^{k},L,v_{\mu_{{L_{2}{({L_{2} - 1})}} + 1}}^{k}} \right)^{T}.\mspace{20mu} S} = \begin{bmatrix}S_{1} & 0_{{L_{1}{({L_{1} - 1})}} \times 1} \\0_{{L_{2}{({L_{2} - 1})}} \times 1} & S_{2} \\{c_{a,1},L,c_{a,M},0_{1 \times {({L_{R} - M})}}} & {c_{b,1},L,c_{b,M},0_{1 \times {({L_{R} - M})}}}\end{bmatrix}}},\mspace{20mu} {q = \begin{pmatrix}0_{{\lbrack{{L_{1}{({L_{1} - 1})}} + {L_{2}{({L_{2} - 1})}}}\rbrack} \times 1} \\{P_{R}/\sigma_{n}^{2}}\end{pmatrix}},{where}}$$\mspace{20mu} {{c_{a,i} = {{\frac{P_{1}}{L_{1}\sigma_{n}^{2}}\lambda_{{R\; 1},i}} + 1}},{c_{b,i} = {{\frac{P_{2}}{L_{2}\sigma_{n}^{2}}\lambda_{{R\; 2},i}} + 1}},{i = 1},L,M,{and}}$  M = min (L₁, L₂, L_(R)).

And

$S_{1} = {\begin{bmatrix}{{Re}\left\{ s_{1} \right\}} \\{{Im}\left\{ s_{1} \right\}}\end{bmatrix} \in {R^{{L_{1}{({L_{1} - 1})}} \times L_{R}}\mspace{14mu} {and}}}$${S_{2} = {\begin{bmatrix}{{Re}\left\{ s_{2} \right\}} \\{{Im}\left\{ s_{2} \right\}}\end{bmatrix} \in R^{{L_{2}{({L_{2} - 1})}} \times L_{R}}}},$

where Re{·} and Im{·} are the functions which take the real andimaginary part of the variables, s₁ and s₂ come from the linearequations

${{\sum\limits_{k = 1}^{L_{R}}{\lambda_{1,k}t_{ik}t_{jk}^{*}}} = 0},{1 \leq i < j \leq L_{1}},{and}$${{\sum\limits_{k = 1}^{L_{R}}{\lambda_{2,k}t_{ki}^{*}t_{kj}}} = 0},{1 \leq i < j \leq L_{2}},$

respectively, i.e.,

${s_{1} = {\begin{pmatrix}{t_{11}t_{21}^{*}} & {t_{12}t_{22}^{*}} & L & {t_{1,L_{g}}t_{2,L_{R}}^{*}} \\{t_{11}t_{31}^{*}} & {t_{12}t_{32}^{*}} & L & {t_{1,L_{R}}t_{3,L_{R}}^{*}} \\M & M & O & M \\{t_{11}t_{L_{1},1}^{*}} & {t_{12}t_{L_{1},2}^{*}} & L & {t_{1,L_{R}}t_{L_{1},L_{R}}^{*}} \\{t_{21}t_{31}^{*}} & {t_{22}t_{32}^{*}} & L & {t_{2,L_{R}}t_{3,L_{R}}^{*}} \\M & M & O & M \\{t_{21}t_{L_{i} - 1}^{*}} & {t_{22}t_{L_{1},2}^{*}} & L & {t_{2,L_{R}}t_{L_{1},L_{R}}^{*}} \\M & M & O & M \\{t_{{L_{1} - 1},1}t_{L_{1},1}^{*}} & {t_{{L_{1} - 1},2}t_{L_{1},2}^{*}} & L & {t_{L_{1} - {1L_{R}}}t_{L_{1},L_{R}}^{*}}\end{pmatrix} \in C^{\frac{L_{1}{({L_{1} - 1})}}{2} \times L_{R}}}},{and}$$\mspace{20mu} {{s_{2} = {{\begin{pmatrix}{t_{11}^{*}t_{12}} & {t_{21}^{*}t_{22}} & L & {t_{L_{R} - 1}^{*}t_{L_{R},2}} \\{t_{11}^{*}t_{13}} & {t_{21}^{*}t_{23}} & L & {t_{L_{R} - 1}^{*}t_{L_{R},3}} \\M & M & O & M \\{t_{11}^{*}t_{1,L_{2}}} & {t_{21}^{*}t_{2,L_{2}}} & L & {t_{L_{R} - 1}^{*}t_{L_{R},2}} \\{t_{12}^{*}t_{13}} & {t_{22}^{*}t_{23}} & L & {t_{L_{R},2}^{*}t_{L_{R},3}} \\M & M & O & M \\{t_{12}^{*}t_{1,L_{2}}} & {t_{22}^{*}t_{2,L_{2}}} & L & {t_{L_{R},2}^{*}t_{L_{R},L_{2}}} \\M & M & O & M \\{t_{1,{L_{2} - 1}}^{*}t_{1,L_{2}}} & {t_{2,{L_{2} - 1}}^{*}t_{2,L_{2}}} & L & {t_{L_{R},{L_{2} - 1}}^{*}t_{L_{R},L_{2}}}\end{pmatrix} \in {C^{\frac{L_{2}{({L_{2} - 1})}}{2} \times L_{R}}.{\nabla^{2}{f_{2}\left( \lambda^{k} \right)}}}} = \begin{pmatrix}\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{1,1}^{2}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,1}}{\partial\lambda_{1,L_{R}}}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,1}}{\partial\lambda_{1,2}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,1}}{\partial\lambda_{2,L_{R}}}} \\M & O & M & M & O & M \\\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,1}}{\partial\lambda_{1,L_{R}}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{1,L_{R}}^{2}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{2,1}}{\partial\lambda_{1,L_{R}}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,L_{R}}}{\partial\lambda_{2,L_{R}}}} \\\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,1}}{\partial\lambda_{2,1}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{2,1}}{\partial\lambda_{1,L_{R}}}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2,1}^{2}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{2,1}}{\partial\lambda_{2,L_{R}}}} \\M & O & M & M & O & M \\\frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,1}}{\partial\lambda_{2,L_{R}}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{1,L_{R}}}{\partial\lambda_{2,L_{R}}}} & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{{\partial\lambda_{2,1}}{\partial\lambda_{2,L_{R}}}} & L & \frac{\partial^{2}{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2,L_{R}}^{2}}\end{pmatrix}}},\mspace{20mu} {and}}$$\mspace{20mu} {{{\nabla{f_{2}\left( \lambda^{k} \right)}} = \left( {\frac{\partial{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{1,1}}L\frac{\partial{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{1.L_{R}}}\frac{\partial{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2,1}}L\frac{\partial{f_{2}\left( \lambda^{k} \right)}}{\partial\lambda_{2,L_{R}}}} \right)^{T}},\mspace{20mu} {\frac{\partial{f_{2}(\lambda)}}{\partial\lambda_{1,j}} = \left\{ {\begin{matrix}{{{\sum\limits_{i = 1}^{M}{{t_{ij}}^{2}\left( {{- m_{1,j}} + m_{2,i}} \right)}} - {c_{a,j}m_{3,j}} + m_{4,j}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ij}}^{2}\left( {{- m_{1,i}} + m_{2,i}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial{f_{2}(\lambda)}}{\partial\lambda_{2,j}} = \left\{ {\begin{matrix}{{{{- c_{b,j}}m_{1,j}} + m_{2,j} + {\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {{- m_{3,i}} + m_{4,i}} \right)}}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ji}}^{2}\left( {{- m_{3,i}} + m_{4,i}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{1,j}^{2}} = \left\{ {\begin{matrix}{{{\sum\limits_{i = 1}^{M}{{t_{ij}}^{2}\left( {m_{1,j}^{2} - m_{2,i}^{2}} \right)}} + {c_{a,j}^{2}m_{3,j}^{2}} - m_{4,j}^{2}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ij}}^{2}\left( {m_{1,j}^{2} - m_{2,i}^{2}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{1,j}^{2}} = {\sum\limits_{i = 1}^{M}{{t_{ij}}^{2}{t_{il}}^{2}\left( {m_{1,i}^{2} - m_{2,i}^{2}} \right)}}},{1 \leq j},{l \leq L_{R}},{j \neq l},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{\partial\lambda_{2,j}^{2}} = \left\{ {\begin{matrix}{{{c_{b,j}^{2}m_{1,j}} - m_{2,j}^{2} + {\sum\limits_{i = 1}^{M}{{t_{ji}}^{4}\left( {m_{3,i}^{2} - m_{4,i}^{2}} \right)}}},{1 \leq j \leq M}} \\{{\sum\limits_{i = 1}^{M}{{t_{ji}}^{4}\left( {m_{3,i}^{2} - m_{4,i}^{2}} \right)}},{{M + 1} \leq j \leq L_{R}}}\end{matrix},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{{\partial\lambda_{2,j}}{\partial\lambda_{2,l}}} = {\sum\limits_{i = 1}^{M}{{t_{ij}}^{2}{t_{il}}^{2}\left( {m_{3,i}^{2} - m_{4,i}^{2}} \right)}}},{1 \leq j},{l \leq L_{R}},\mspace{20mu} {\frac{\partial^{2}{f_{2}(\lambda)}}{{\partial\lambda_{1,j}}{\partial\lambda_{2,l}}} = \left\{ {\begin{matrix}{{{t_{lj}}^{2}\left( {{c_{b,i}m_{1,i}^{2}} - m_{2,i}^{2} + {c_{a,i}m_{3,i}^{2}} - m_{4,i}^{2}} \right)},{1 \leq j},{l \leq M}} \\{{{t_{lj}}^{2}\left( {{c_{a,i}m_{3,i}^{2}} - m_{4,i}^{2}} \right)},{1 \leq j \leq M},{{M + 1} \leq l \leq L_{R}}} \\{{{t_{lj}}^{2}\left( {{c_{b,i}m_{1,j}^{2}} - m_{2,i}^{2}} \right)},{{M + 1} \leq j \leq N_{R}},{1 \leq l \leq M},} \\{0,{{M + 1} \leq j},{l \leq L_{R}}}\end{matrix},\mspace{20mu} {m_{1,i} = \frac{\lambda_{{1R},i}}{1 + {\sum\limits_{k = 1}^{L_{R}}{\lambda_{1,k}{t_{ik}}^{2}\lambda_{{1\; R},i}}} + {\lambda_{2,i}c_{b,i}\lambda_{{1R},i}}}},\mspace{20mu} {m_{2,i} = \frac{\lambda_{{1R},i}}{1 + {\sum\limits_{k = 1}^{L_{R}}{\lambda_{1,k}{t_{ik}}^{2}\lambda_{{1R},i}}} + {\lambda_{2,i}\lambda_{{1R},i}}}},\mspace{20mu} {m_{3,i} = \frac{\lambda_{{2R},i}}{1 + {\lambda_{1}c_{a,i}\lambda_{{2R},i}} + {\sum\limits_{k = 1}^{L_{R}}{\lambda_{2,k}{t_{ki}}^{2}\lambda_{{2R},i}}}}},\mspace{20mu} {m_{4,i} = {\frac{\lambda_{{2R},i}}{1 + {\lambda_{1}\lambda_{{2R},i}} + {\sum\limits_{k = 1}^{L_{R}}{\lambda_{2,k}{t_{ki}}^{2}\lambda_{{2R},i}}}}.}}} \right.}} \right.}} \right.}} \right.}} \right.}}$

Denote

${M_{1} = {{H_{R\; 2}\left\lbrack {I_{L_{2}} + {{H_{12}^{H}\left( {I_{L_{1}} + {\frac{P_{2}}{L_{2}\sigma_{n}^{2}}H_{12}H_{12}^{H}}} \right)}^{- 1}\frac{P_{2}}{L_{2}\sigma_{n}^{2}}H_{12}}} \right\rbrack}H_{R\; 2}^{H}}},{and}$$M_{2} = {{H_{R\; 1}\left\lbrack {I_{L_{1}} + {{H_{21}^{H}\left( {I_{L_{2}} + {\frac{P_{1}}{L_{1}\sigma_{n}^{2}}H_{21}H_{21}^{H}}} \right)}^{- 1}\frac{P_{1}}{L_{1}\sigma_{n}^{2}}H_{12}}} \right\rbrack}{H_{R\; 1}^{H}.}}$

They are both Hermitian, and M₁=U_(M) ₁ A_(M) ₁ U_(M) ₁ ^(H), andM₂=U_(M) ₂ A_(M) ₂ U_(M) ₂ ^(H), with U_(M) ₁ , U_(M) ₂ ∈C^(L) ^(×L)^(R) are unitary matrices, and A_(M) ₁ =diag {λ_(M) _(i) _(,i)} andA_(M) ₂ =diag{λ_(M) ₂ ^(,i)} are diagonal matrices.

Let W₁=V_(2R){tilde over (W)}₁U_(M) ₂ ^(H) and W₂=V_(1R){tilde over(W)}₂U_(M) ₁ ^(H). The problem to determine {tilde over (W)}₁, {tildeover (W)}₂∈C^(L) ^(R) ^(×L) ^(R) , when the CSI of the direct linksbetween Node₁ and Node₂ is available, the RN is the same form as abovewhen the CSI for the direct links between Node₁ and Node₂ is notavailable and the iterative approach of finding the solution is almostthe same as the three cases above by just replacing λ_(R2,i) andλ_(R1,i) by λ_(M) ₁ ^(,i) and λ_(M) ₂ ^(,i), respectively.

In the design problem, both nodes (source and destination) send out(transmit, communicate, forward) training sequences to the RN, so thatthe RN can estimate the incoming channels. The RN also needs to send out(transmit, communicate, forward) training sequence(s) for the nodes(source and destination) to estimate the channels (links) from the RN toeach of them and also transmit (send out, forward, communicate)information regarding the pre-coding matrices the RN uses.

Two basic data frame structures are proposed for use in the presentinvention:

(1) The RN applies the pre-coding matrices to the training sequences itreceives and forwards them, as in FIG. 4( a) and FIG. 5( a).

(2) The RN estimates the incoming channel matrices, multiplies them withthe pre-coding matrices, quantizes the resulting matrices, and feedsthem back. It also sends out its own training sequence to the nodes(source and destination) to estimate the channel state (channelconditions) from the RN to them, as in FIG. 4( b) and FIG. 5( b). Stillthere are other channel estimations that need to be done, such as theCSI of the outgoing channels from RN to nodes (source and destination)at the RN and the CSI of the direct links at the RN. They are done byother frames, such as control frames, etc.

Referring again to FIG. 3 a, which is a block diagram of the operationof an exemplary embodiment of the transmission side of the Network-CodedAmplify-and-Forward relaying scheme of the present invention. Data S₁ istransmitted (communicated) from Node₁ as signal X₁ to the RN in thefirst time slot and data S₂ is transmitted (communicated) from Node₂ assignal X₂ to the RN in the second time slot. The RN then pre-codes,mixes (combines) and multicasts (broadcasts) the pre-coded mixed data(X_(R)) for receipt by both Node₁ and Node₂. It goes without saying thatthe pre-coded mixed (combined) data is modulated.

Referring again to FIG. 3 b, which is a block diagram of the operationof an exemplary embodiment of the reception side of the Network-CodedAmplify-and-Forward relaying scheme of the present invention from theperspective of Node₁. Node₁ receives signal Y₁₂ from Node₂ and signalY_(1R) from the RN and performs joint network and channel decoding. Itgoes without saying that the decoded data is demodulated.

Referring again to FIG. 3 c, which is a block diagram of the operationof an exemplary embodiment of the reception side of the Network-CodedAmplify-and-Forward relaying scheme of the present invention from theperspective of Node₂. Node₂ receives signal Y₂₁ from Node₁ and signalY_(2R) from the RN and performs joint network and channel decoding. Itgoes without saying that the decoded data is demodulated.

That is, from the standpoint (perspective) of a node that holds(possesses) the transmission opportunity (TXOP), the node includes meansfor transmitting (sending out, communicating) a first signal having dataand means for receiving a second signal. The transmitting and receivingmeans may be a transceiver or a separate transmitter and a separatereceiver or any equivalent means. The node also has means for jointlydecoding the second signal by removing (subtracting off) a trainingsequence and the first signal. Optionally, the node also includes meansfor decoding the second signal by accounting for the application of afirst beamforming matrix to the data of the first signal and theapplication of a second beamforming matrix to the data of the secondsignal.

From the standpoint (perspective) of a relay node (RN), the RN includesmeans for receiving a first signal in a first time slot of a firstchannel, means for receiving a second signal in a second time slot of asecond channel, means for determining a first pre-coding matrix, meansfor determining a second pre-coding matrix, wherein the first pre-codingmatrix and the second pre-coding matrix maximize a joint channelcapacity of the first channel and second channel respectively, means forapplying the first pre-coding matrix to first data, means for applyingthe second pre-coding matrix to second data, means for generating athird signal by mixing (combining) the pre-coded first data and thepre-coded second data and means for transmitting the third signal. Thetransmitting and receiving means may be a transceiver or a separatetransmitter and a separate receiver or any equivalent means. The RN alsoincludes means for generating a first estimated channel matrix for thefirst channel and a second estimated channel matrix for the secondchannel, means for inserting the first pre-coding matrix times the firstestimated channel matrix and the second pre-coding matrix times thesecond estimated channel matrix between a training sequence of the thirdsignal and means for quantizing the first pre-coding matrix times thefirst estimated channel matrix and means for quantizing the secondpre-coding matrix times the second estimated channel matrix before themeans for inserting is executed. Optionally, the RN also includes meansfor determining a first beamforming matrix, means for determining asecond beamforming matrix and means for applying the first beamformingmatrix to the first data and applying the second beamforming matrix tothe second data before generating the third signal. The RN alsooptionally includes means for applying the first beamforming matrix tothe first pre-coded matrix times the first estimated channel matrix andmeans for applying the second beamforming matrix to the second pre-codedmatrix times the second estimated channel matrix before inserting thefirst pre-coded matrix times the first estimated channel matrix and thesecond pre-coded matrix times the second estimated channel matrixbetween the training sequence of the third signal.

Referring again to FIG. 4, which is collectively simplified exemplaryframe structures for the Network-Coded Amplify-and-Forward relayingscheme without beamforming at source nodes of the present invention.FIG. 4 a shows the case where training sequences are sent from thesource nodes. The RN sends out training sequences but the trainingsequences sent by the RN are effectively the training sequences that theRN received from Node₁ and Node₂ copied and returned to the source nodes(Node₁ and Node₂). In the third stage the RN transmits the two trainingsequences and the mixed data X_(R) to Node_(i). Pre-coding matrices areapplied to both the training sequences and to the mixed data X_(R).Pre-coding matrix W₁ is applied to T₁ (the first training sequence) andpre-coding matrix W₂ is applied to T₂ (the second training sequence).Pre-coding matrix W₁ is also applied to H_(R1)X₁ of the mixed signalX_(R) and W₂ is applied to H_(R2)X₂ of the mixed signal X_(R). Note thatH_(R1)X₁ is the desired signal that the RN received from Node₁ (Node₁sent X₁). Also note that H_(R2)X₂ is the desired signal that the RNreceived from Node₂ (Node₂ sent X₂). In FIG. 4 b, each node sends out(transmits, communicates) its own training sequence and the RN alsosends out (transmits, communicates) pre-coding matrices W₁ and W₂ inaddition to the mixed data X_(R). No pre-coding matrix is applied to thetraining sequence T_(R) sent out (transmitted, communicated) by the RN.

Referring again to FIG. 5, which is collectively simplified exemplaryframe structures for the Network-Coded Amplify-and-Forward relayingscheme with beamforming at source nodes of the present invention. FIG. 5a shows the case where training sequences are sent from the sourcenodes. The RN sends out training sequences but the training sequencessent by the RN are effectively the training sequences that the RNreceived from Node₁ and Node₂ copied and returned to the source nodes(Node₁ and Node₂). In the third stage the RN transmits the two trainingsequences and the mixed data X_(R) to Node_(i). Pre-coding matrices areapplied to both the training sequences and to the mixed data X_(R).Pre-coding matrix W₁ is applied to T₁ (the first training sequence) andpre-coding matrix W₂ is applied to T₂ (the second training sequence).Pre-coding matrix W₁ is also applied to H_(R1)X₁ of the mixed signalX_(R) and W2 is applied to H_(R2)X₂ of the mixed signal X_(R). Note thatH_(R1)X₁ is the desired signal that the RN received from Node₁ (Node₁sent X₁). Also note that H_(R2)X₂ is the desired signal that the RNreceived from Node₂ (Node₂ sent X₂). In addition to applying thepre-coding matrices to the training sequences and the mixed data,beamforming matrix Q₁ is applied to training sequence T₁ and beamformingmatrix Q₂ is applied to training sequence T₂. Beamforming matrix Q₁ isalso applied to X₁ of the mixed signal X_(R) and beamforming matrix Q₂is applied to X₂ of mixed signal X_(R). In FIG. 5 b, each node sends out(transmits, communicates) its own training sequence and the RN alsosends out (transmits, communicates) pre-coding matrices W₁ and W₂ inaddition to the mixed signal X_(R). No pre-coding matrix is applied tothe training sequence TR sent out (transmitted, communicated) by the RN.Pre-coding matrices are applied to the mixed signal X_(R). Pre-codingmatrix W₁ is also applied to H_(R1)X₁ of the mixed signal X_(R) and W2is applied to H_(R2)X₂ of the mixed signal X_(R). Note that H_(R1)X₁ isthe desired signal that the RN received from Node₁ (Node₁ sent X₁). Alsonote that H_(R2)X₂ is the desired signal that the RN received from Node₂(Node₂ sent X₂). In addition to applying the pre-coding matrices tomixed signal, beamforming matrix Q₁ is also applied to X₁ of the mixedsignal X_(R) and beamforming matrix Q₂ is applied to X₂ of mixed signalX_(R).

Referring to FIG. 6, which is a flowchart of an exemplary embodiment ofthe present invention from the perspective of a node (source anddestination), at 605 the node transmits a first signal (message)including data. At 610, the node receives a second signal (message)including data. At 615, the node jointly decodes the second signal(message) which was a combined signal (message) including both the firstdata transmitted by the node and third data transmitted by another node(destination node for the first data and source node for the third data)by removing (subtracting out) the first data and the training sequencesince both the training sequence and the first data are known as well asthe pre-coding matrix that was applied to the first data.

Referring to FIG. 7, which is a flowchart of an exemplary embodiment ofthe present invention from the perspective of a relay node, at 705 therelay node receives a first signal (message) in a first time slot of afirst channel. The first signal includes first data and a first trainingsequence. At 710 the relay node receives a second signal (message) in asecond time slot of a second channel. The second signal includes seconddata and a second training sequence. At 715 the relay node determines afirst pre-coding matrix to maximize the joint capacity of the channels.At 720 the relay node determines a second pre-coding matrix to maximizethe joint capacity of the channels. At 725 the relay node applies thefirst pre-coding matrix to first data. At 730 the relay node applies thesecond pre-coding matrix to second data. At 735 the relay node generatesa third signal by mixing (combining) the pre-coded first data and thepre-coded second data. At 740 the relay node transmits (multicasts,broadcasts, communicates, sends out) the third data over (on) both thefirst and the second channels.

It is to be understood that the present invention may be implemented invarious forms of hardware, software, firmware, special purposeprocessors, or a combination thereof. Preferably, the present inventionis implemented as a combination of hardware and software. Moreover, thesoftware is preferably implemented as an application program tangiblyembodied on a program storage device. The application program may beuploaded to, and executed by, a machine comprising any suitablearchitecture. Preferably, the machine is implemented on a computerplatform having hardware such as one or more central processing units(CPU), a random access memory (RAM), and input/output (I/O)interface(s). The computer platform also includes an operating systemand microinstruction code. The various processes and functions describedherein may either be part of the microinstruction code or part of theapplication program (or a combination thereof), which is executed viathe operating system. In addition, various other peripheral devices maybe connected to the computer platform such as an additional data storagedevice and a printing device.

It is to be further understood that, because some of the constituentsystem components and method steps depicted in the accompanying figuresare preferably implemented in software, the actual connections betweenthe system components (or the process steps) may differ depending uponthe manner in which the present invention is programmed. Given theteachings herein, one of ordinary skill in the related art will be ableto contemplate these and similar implementations or configurations ofthe present invention.

1-24. (canceled)
 25. A method, said method comprising: transmitting afirst signal, wherein said first signal includes first data; receiving asecond signal, wherein said second signal includes pre-coded datagenerated in a relay node by applying a first pre-coding matrix to saidfirst data to maximize a joint channel capacity of a first channel andapplying a second pre-coding matrix to second data to maximize a jointchannel capacity of a second channel, said first pre-coding matrix andsaid second pre-coding matrix having been determined by the relay node;and decoding said second signal by removing said first signal. 26-29.(canceled)
 30. The method according to claim 25, further comprising afirst result generated by said first pre-coding matrix times a firstestimated channel matrix and a second result generated by said secondpre-coding matrix times a second estimated channel matrix, said firstresult and said second result included between a first training sequenceand data of said second signal.
 31. The method according to claim 25,wherein a first beamforming matrix was applied to said first pre-codingmatrix and said first data and a second beamforming matrix was appliedto said second pre-coding matrix and said second data.
 32. An apparatuscomprising: a transmitter, said transmitter transmitting a first signal,wherein said first signal includes first data; a receiver, said receiverreceiving a second signal, wherein said second signal includes pre-codeddata generated in a relay node by applying a first pre-coding matrix tosaid first data to maximize a joint channel capacity of a first channeland applying a second pre-coding matrix to second data to maximize ajoint channel capacity of a second channel, said first pre-coding matrixand said second pre-coding matrix having been determined by the relaynode; and a processor, said processor decoding said second signal byremoving said first signal. 33-37. (canceled)
 38. The apparatusaccording to claim 32, wherein a first beamforming matrix was applied tosaid first pre-coding matrix and said first data and a secondbeamforming matrix was applied to said second pre-coding matrix and saidsecond data.
 39. A relay node comprising: a receiver, said receiverreceiving a first signal including first data in a first time slot of afirst channel; said receiver, receiving a second signal including seconddata in a second time slot of second channel, wherein said first signaland said second signal are exchanged between a source node and adestination node communicating with each other via said relay node; aprocessor, said processor determining a first pre-coding matrix tomaximize a joint channel capacity of the first channel, said processorin communication with said receiver; said processor, determining asecond pre-coding matrix to maximize a joint channel capacity of thesecond channel; said processor, applying said first pre-coding matrix tosaid first data to produce pre-coded first data, wherein said applyingis accomplished by matrix multiplication; said processor, applying saidsecond pre-coding matrix to said second data to produce second pre-codeddata, wherein said applying is accomplished by matrix multiplication;said processor, generating a third signal by combining said pre-codedfirst data and said pre-coded second data, wherein said combining isaccomplished by matrix addition; and a transmitter, said transmittermulticasting said third signal on said first channel and on said secondchannel, wherein said relay node is in a bi-directional communicationssystem, said transmitter is in communication with said processor. 40.The relay node according to claim 39, wherein said third signal furthercomprises a training sequence prepended to said third signal.
 41. Therelay node according to claim 40, wherein said processor furthergenerates a first estimated channel matrix for said first channel and asecond estimated channel matrix for said second channel.
 42. The relaynode according to claim 41, wherein said processor further inserts saidfirst pre-coding matrix times said first estimated channel matrix andsaid second pre-coding matrix times said second estimated channel matrixbetween said training sequence of said third signal.
 43. The relay nodeaccording to claim 42, wherein said processor further quantizes saidfirst pre-coding matrix times said first estimated channel matrix andquantizes said second pre-coding matrix times said second estimatedchannel matrix before performing said inserting.
 44. The relay nodeaccording to claim 42, wherein said training sequence includes a firsttraining sequence and a second training sequence, said first trainingsequence is for said first channel and said second training sequence isfor said second channel.
 45. The relay node according to claim 39,wherein said first signal includes a training sequence and data.
 46. Therelay node according to claim 39, wherein said second signal includes atraining sequence and data.
 47. The relay node according to claim 41,further comprising: said processor, determining a first beamformingmatrix; said processor, determining a second beamforming matrix; andsaid processor, applying said first beamforming matrix to said firstdata and applying said second beamforming matrix to said second databefore generating said third signal.
 48. The relay node according toclaim 47, further comprising said processor applying said firstbeamforming matrix to said first pre-coded matrix times said firstestimated channel matrix and applying said second beamforming matrix tosaid second pre-coded matrix times said second estimated channel matrixbefore inserting said first pre-coded matrix times said first estimatedchannel matrix and said second pre-coded matrix times said secondestimated channel matrix between said training sequence of said thirdsignal.
 49. The relay node according to claim 39, wherein said firstchannel and said second channel are orthogonal.
 50. The relay nodeaccording to claim 39, wherein said second time slot is a reversedirection granted by a node which used said first time slot.